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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2309.07031 |
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| _version_ | 1866929725094494208 |
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| author | Liu, Tianle Austern, Morgane |
| author_facet | Liu, Tianle Austern, Morgane |
| contents | Recent progress has been made in establishing normal approximation bounds in terms of the Wasserstein-$p$ distance for i.i.d. and locally dependent random variables. However, for $p > 1$, no such results have been demonstrated for dependent variables under $α$-mixing conditions. In this paper, we extend the Wasserstein-$p$ bounds to $α$-mixing random fields. We show that, under appropriate conditions, the rescaled average of random fields converges to the standard normal distribution in the Wasserstein-$p$ distance at a rate of $O(|T|^{-β})$, where $|T|$ is the size of the index set, and $β\in (0, 1/2]$ depends on $p$, the dimension $d$ of the random fields, and the decay rate of the $α$-mixing coefficients. Notably, $β= 1/2$ is achievable if the mixing coefficients decay at a sufficiently fast polynomial rate. Our results are derived through a carefully constructed cumulant-based Edgeworth expansion and an adaptation of recent developments in Stein's method. Additionally, we introduce a novel constructive graph approach that leverages combinatorial techniques to establish the desired expansion for general dependent variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07031 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields Liu, Tianle Austern, Morgane Probability 60F05 Recent progress has been made in establishing normal approximation bounds in terms of the Wasserstein-$p$ distance for i.i.d. and locally dependent random variables. However, for $p > 1$, no such results have been demonstrated for dependent variables under $α$-mixing conditions. In this paper, we extend the Wasserstein-$p$ bounds to $α$-mixing random fields. We show that, under appropriate conditions, the rescaled average of random fields converges to the standard normal distribution in the Wasserstein-$p$ distance at a rate of $O(|T|^{-β})$, where $|T|$ is the size of the index set, and $β\in (0, 1/2]$ depends on $p$, the dimension $d$ of the random fields, and the decay rate of the $α$-mixing coefficients. Notably, $β= 1/2$ is achievable if the mixing coefficients decay at a sufficiently fast polynomial rate. Our results are derived through a carefully constructed cumulant-based Edgeworth expansion and an adaptation of recent developments in Stein's method. Additionally, we introduce a novel constructive graph approach that leverages combinatorial techniques to establish the desired expansion for general dependent variables. |
| title | Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields |
| topic | Probability 60F05 |
| url | https://arxiv.org/abs/2309.07031 |